MathDB

pp-parity conjecture for elliptic curves

From papers

Let EE be an elliptic curve over a number field KK, let pp be a prime, and let rk⁡pE/K\operatorname{rk}_p E/K denote the p∞p^\infty-Selmer rank, namely the Mordell–Weil rank plus the number of copies of Qp/Zp\mathbb{Q}_p/\mathbb{Z}_p occurring in the Tate–Shafarevich group \Sha(E/K)\Sha(E/K).

pp-parity conjecture. The p∞p^\infty-Selmer rank rk⁡pE/K\operatorname{rk}_p E/K is even if and only if w(E/K)=1w(E/K)=1.

This is the Selmer-rank formulation of the parity conjecture. The source presents it as a conjectural statement, while noting that the paper proves the corresponding assertion for elliptic curves over Q\mathbb{Q} for all primes pp.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The pp-parity conjecture for elliptic curves

    Let EE be an elliptic curve over a number field KK, and let Sel⁡p∞(E/K)\operatorname{Sel}_{p^\infty}(E/K) be its p∞p^\infty-Selmer group. Write rk⁡p∞(E/K)\operatorname{rk}_{p^\infty}(E/K) for its Zp\mathbb Z_p-corank, and let w(E/K)∈{±1}w(E/K)\in\{\pm1\} be the global root number.

    pp-parity conjecture.

    (−1)rk⁡p∞(E/K)=w(E/K).(-1)^{\operatorname{rk}_{p^\infty}(E/K)}=w(E/K).

    This is a Selmer-theoretic refinement of the parity conjecture. The paper proves this in several cases, but the general assertion is not established.

    source: Tim Dokchitser and Vladimir Dokchitser, “Root numbers and parity of ranks of elliptic curves”, arXiv:0906.1815 (2009).

Sources & referencesView supporting material

Primary source

Tim Dokchitser and Vladimir Dokchitser, “On the Birch-Swinnerton-Dyer quotients modulo squares”, arXiv:math/0610290 (2008).

elliptic-curvesparity-conjectureroot-numbersselmer-groups

Solutions 0

No solutions have been posted yet.